In algebraic geometry books, it seems that the motivation of algebraic geometry is to understand the set of solutions to systems of polynomials.
But, from what I have gathered, Decartes introduced coordinates to study geometry, rather than using geometry to study algebra. I have some intuitive idea to what this could mean. For example, a circle, which was used extensively by Euclid, may be represented by a quadratic formula. From this one could considered the intersection of circles as solving a system of two quadratic polynomials. I have also heard that Grothendieck attempted to study geometry by only using algebraic geometric techniques.
Beyong this elementary intuition, does algebraic geometry say more about geometry? Is algebraic geometry used for more than as a basic computational tool in geometry?